How can I evenly but randomly distrybute particles in a mesh?
The problem is due to the uneven geometry interfering with the distribution of the particles. Add a Remesh modifier just before the Particle System in the modifier stack and, as well as enabling the Even Distribution (as mentioned by @LukeD), ensure to enable Use Modifier Stack. This should result in an even distribution of particles throughout the volume.

Without Remesh :


With Remesh :